![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pi1blem | Structured version Visualization version GIF version |
Description: Lemma for pi1buni 24788. (Contributed by Mario Carneiro, 10-Jul-2015.) |
Ref | Expression |
---|---|
pi1val.g | β’ πΊ = (π½ Ο1 π) |
pi1val.1 | β’ (π β π½ β (TopOnβπ)) |
pi1val.2 | β’ (π β π β π) |
pi1val.o | β’ π = (π½ Ξ©1 π) |
pi1bas.b | β’ (π β π΅ = (BaseβπΊ)) |
pi1bas.k | β’ (π β πΎ = (Baseβπ)) |
Ref | Expression |
---|---|
pi1blem | β’ (π β ((( βphβπ½) β πΎ) β πΎ β§ πΎ β (II Cn π½))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3477 | . . . . 5 β’ π₯ β V | |
2 | 1 | elima 6064 | . . . 4 β’ (π₯ β (( βphβπ½) β πΎ) β βπ¦ β πΎ π¦( βphβπ½)π₯) |
3 | simpr 484 | . . . . . . . . 9 β’ ((π β§ π¦( βphβπ½)π₯) β π¦( βphβπ½)π₯) | |
4 | isphtpc 24741 | . . . . . . . . 9 β’ (π¦( βphβπ½)π₯ β (π¦ β (II Cn π½) β§ π₯ β (II Cn π½) β§ (π¦(PHtpyβπ½)π₯) β β )) | |
5 | 3, 4 | sylib 217 | . . . . . . . 8 β’ ((π β§ π¦( βphβπ½)π₯) β (π¦ β (II Cn π½) β§ π₯ β (II Cn π½) β§ (π¦(PHtpyβπ½)π₯) β β )) |
6 | 5 | adantrl 713 | . . . . . . 7 β’ ((π β§ (π¦ β πΎ β§ π¦( βphβπ½)π₯)) β (π¦ β (II Cn π½) β§ π₯ β (II Cn π½) β§ (π¦(PHtpyβπ½)π₯) β β )) |
7 | 6 | simp2d 1142 | . . . . . 6 β’ ((π β§ (π¦ β πΎ β§ π¦( βphβπ½)π₯)) β π₯ β (II Cn π½)) |
8 | phtpc01 24743 | . . . . . . . . 9 β’ (π¦( βphβπ½)π₯ β ((π¦β0) = (π₯β0) β§ (π¦β1) = (π₯β1))) | |
9 | 8 | ad2antll 726 | . . . . . . . 8 β’ ((π β§ (π¦ β πΎ β§ π¦( βphβπ½)π₯)) β ((π¦β0) = (π₯β0) β§ (π¦β1) = (π₯β1))) |
10 | 9 | simpld 494 | . . . . . . 7 β’ ((π β§ (π¦ β πΎ β§ π¦( βphβπ½)π₯)) β (π¦β0) = (π₯β0)) |
11 | pi1val.o | . . . . . . . . . . 11 β’ π = (π½ Ξ©1 π) | |
12 | pi1val.1 | . . . . . . . . . . 11 β’ (π β π½ β (TopOnβπ)) | |
13 | pi1val.2 | . . . . . . . . . . 11 β’ (π β π β π) | |
14 | pi1bas.k | . . . . . . . . . . 11 β’ (π β πΎ = (Baseβπ)) | |
15 | 11, 12, 13, 14 | om1elbas 24780 | . . . . . . . . . 10 β’ (π β (π¦ β πΎ β (π¦ β (II Cn π½) β§ (π¦β0) = π β§ (π¦β1) = π))) |
16 | 15 | biimpa 476 | . . . . . . . . 9 β’ ((π β§ π¦ β πΎ) β (π¦ β (II Cn π½) β§ (π¦β0) = π β§ (π¦β1) = π)) |
17 | 16 | adantrr 714 | . . . . . . . 8 β’ ((π β§ (π¦ β πΎ β§ π¦( βphβπ½)π₯)) β (π¦ β (II Cn π½) β§ (π¦β0) = π β§ (π¦β1) = π)) |
18 | 17 | simp2d 1142 | . . . . . . 7 β’ ((π β§ (π¦ β πΎ β§ π¦( βphβπ½)π₯)) β (π¦β0) = π) |
19 | 10, 18 | eqtr3d 2773 | . . . . . 6 β’ ((π β§ (π¦ β πΎ β§ π¦( βphβπ½)π₯)) β (π₯β0) = π) |
20 | 9 | simprd 495 | . . . . . . 7 β’ ((π β§ (π¦ β πΎ β§ π¦( βphβπ½)π₯)) β (π¦β1) = (π₯β1)) |
21 | 17 | simp3d 1143 | . . . . . . 7 β’ ((π β§ (π¦ β πΎ β§ π¦( βphβπ½)π₯)) β (π¦β1) = π) |
22 | 20, 21 | eqtr3d 2773 | . . . . . 6 β’ ((π β§ (π¦ β πΎ β§ π¦( βphβπ½)π₯)) β (π₯β1) = π) |
23 | 11, 12, 13, 14 | om1elbas 24780 | . . . . . . 7 β’ (π β (π₯ β πΎ β (π₯ β (II Cn π½) β§ (π₯β0) = π β§ (π₯β1) = π))) |
24 | 23 | adantr 480 | . . . . . 6 β’ ((π β§ (π¦ β πΎ β§ π¦( βphβπ½)π₯)) β (π₯ β πΎ β (π₯ β (II Cn π½) β§ (π₯β0) = π β§ (π₯β1) = π))) |
25 | 7, 19, 22, 24 | mpbir3and 1341 | . . . . 5 β’ ((π β§ (π¦ β πΎ β§ π¦( βphβπ½)π₯)) β π₯ β πΎ) |
26 | 25 | rexlimdvaa 3155 | . . . 4 β’ (π β (βπ¦ β πΎ π¦( βphβπ½)π₯ β π₯ β πΎ)) |
27 | 2, 26 | biimtrid 241 | . . 3 β’ (π β (π₯ β (( βphβπ½) β πΎ) β π₯ β πΎ)) |
28 | 27 | ssrdv 3988 | . 2 β’ (π β (( βphβπ½) β πΎ) β πΎ) |
29 | simp1 1135 | . . . 4 β’ ((π₯ β (II Cn π½) β§ (π₯β0) = π β§ (π₯β1) = π) β π₯ β (II Cn π½)) | |
30 | 23, 29 | syl6bi 253 | . . 3 β’ (π β (π₯ β πΎ β π₯ β (II Cn π½))) |
31 | 30 | ssrdv 3988 | . 2 β’ (π β πΎ β (II Cn π½)) |
32 | 28, 31 | jca 511 | 1 β’ (π β ((( βphβπ½) β πΎ) β πΎ β§ πΎ β (II Cn π½))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2939 βwrex 3069 β wss 3948 β c0 4322 class class class wbr 5148 β cima 5679 βcfv 6543 (class class class)co 7412 0cc0 11113 1c1 11114 Basecbs 17149 TopOnctopon 22633 Cn ccn 22949 IIcii 24616 PHtpycphtpy 24715 βphcphtpc 24716 Ξ©1 comi 24749 Ο1 cpi1 24751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9440 df-inf 9441 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-icc 13336 df-fz 13490 df-seq 13972 df-exp 14033 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-struct 17085 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-tset 17221 df-topgen 17394 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-top 22617 df-topon 22634 df-bases 22670 df-cn 22952 df-ii 24618 df-htpy 24717 df-phtpy 24718 df-phtpc 24739 df-om1 24754 |
This theorem is referenced by: pi1buni 24788 pi1bas3 24791 pi1addf 24795 pi1addval 24796 pi1grplem 24797 |
Copyright terms: Public domain | W3C validator |